safesphere responded to my question by saying that "time and energy are Fourier conjugates (or more generally, spacetime and energy-momentum) and cannot exist in the physical reality without each other. In other words, GR states that spacetime is the field produced by matter just like the electromagnetic field is produced by charges."
My response is the following:
Spacetime and energy-momentum are not Fourier conjugates. In Newton mechanics they are topological duals (flows and generators).
This connection is different in curved spacetime. In curved spacetime the Fourier transform is not definable. This is the reason in mathematics some differential equations are perfectly understood in a flat space are not nearly understood even in simple curved spaces.
In GR it is not even possible to formulate the claim that spacetime and energy-momentum are Fourier conjugates.
There are many more things one could say here. There is a connection between spacetime and energy-momentum (this is the reason why energy is not conserved in GR). But only quantum observables in flat spacetime are connected via some kind Fourier transform. But even this connection is wrong for photons because they have no sharp spacetime observable but an energy-momentum observable.
As I said, I could say many more things here. But bottom line is, that in GR energy-momentum and spacetime are not Fourier conjugates.